To solve the given expression
[tex]\[
\frac{\left(4 m^2 n\right)^2}{2 m^5 n},
\][/tex]
we need to simplify it step by step.
### Step 1: Simplify the numerator
First, let's simplify the numerator [tex]\((4 m^2 n)^2\)[/tex].
[tex]\[
(4 m^2 n)^2 = 4^2 \cdot (m^2)^2 \cdot n^2 = 16 \cdot m^4 \cdot n^2
\][/tex]
Thus, the numerator simplifies to [tex]\( 16 m^4 n^2 \)[/tex].
### Step 2: Rewrite the expression with the simplified numerator
Now, substitute the simplified numerator back into the expression:
[tex]\[
\frac{16 m^4 n^2}{2 m^5 n}
\][/tex]
### Step 3: Simplify the expression
Next, we need to divide the numerator by the denominator. Let's break it down:
[tex]\[
\frac{16 m^4 n^2}{2 m^5 n} = \frac{16}{2} \cdot \frac{m^4}{m^5} \cdot \frac{n^2}{n}
\][/tex]
Simplify each part separately:
- [tex]\(\frac{16}{2} = 8\)[/tex]
- [tex]\(\frac{m^4}{m^5} = m^{4-5} = m^{-1}\)[/tex]
- [tex]\(\frac{n^2}{n} = n^{2-1} = n\)[/tex]
### Step 4: Combine the simplified components
Putting it all together:
[tex]\[
8 \cdot m^{-1} \cdot n = 8 m^{-1} n
\][/tex]
### Conclusion
The equivalent expression is [tex]\(8 m^{-1} n\)[/tex], which corresponds to option B.
So, the correct answer is:
[tex]\[
\boxed{B. \, 8 m^{-1} n}
\][/tex]
